Correlation
Correlation can be loosely defined as the mutual relation between two or more variables. This is captured in a quantitative way in statistics by a measure defined as the correlation coefficient, usually represented by the letter “r ”. The linear correlation coefficient is used to examine whether there is any evidence of a linear relationship between two variables and defines two qualities:
Nature of relationship. A positive correlation coefficient means that the two variables tend to move in the same direction. If it is negative it implies that the relationship is inverse and that when the value of one variable rises the value of the other tends to fall. Strength of relationship. The value of correlation coefficient provides a measure of the strength of the relationship, if any exists. The correlation coefficient cannot be greater than +1 or less than –1:
Perfect linear relationship. A correlation coefficient of +1 means that there is a perfect linear relationship. If one variable rises then so does the other and the ratio of the rise or fall remains constant. If, for example, the price of one bond increases then the price of the other bond always rises. If the increase in price of the first bond for a given fall in yields is 1% and the price of the second rises by 2% then if the first bond falls in value by 3% the value of the second bond will fall by 6%. Perfect inverse linear relationship. A correlation coefficient of –1 means there is a perfect inverse linear relationship. In the above example a 1% rise in the value of the first bond would be associated with a 2% fall in the value of the second bond. If the first bond falls by 3% the value of the second bond increases by 6%. Independent variables. A correlation coefficient of zero suggests that no linear relationship exists and that the variables move independently of one another.
Note that these are expressed in absolute terms. A “trend line” has been added to the scatter graph. This is defined as the line of best fit and is the line about which the variation of values is minimized. Credit spreads and long-term risk-free rates appear to have an inverse relationship. When long-term risk-free rates rise credit spreads tend to narrow and when yields fall tend to widen.
The variation around the line of best fit is far greater than it is for the plot of change in credit spreads versus changes in long-term yields. There is a weak positive correlation.
In more technical terms the square of the correlation coefficient r2 defines the “goodness of fit” of the trend line. It measures the proportion of the values that can be explained by the inferred linear relationship. Methods exist to assess whether the results are statistically significant, whether it is likely that a linear relationship does in fact exist or whether the results can be explained by chance alone.
A widely used rule of thumb is that if the value for r2 is less than 0.4 (r = 0.63) the evidence for any linear relationship is very thin. A large value for r is not in itself proof that a relationship exists and a useful simple check is to calculate r over a range of time frames to determine whether its value remains stable.
Did you enjoy this post? Why not leave a comment below
Comments
No comments yet.
Sorry, the comment form is closed at this time.